Pedernales Falls State Park. This is a beautiful park only about 30 minutes away. There are hiking trails, a beautiful falls area, springs and the river itself. There are at least 3 swimming areas: Bee Creek (explained elsewhere), the main swimming area, which has (usually) gentle, rocky rapids and the youth-group area which has a sandy beach, deep, still water and a rope swing.
Take 290 West for about 30 miles and turn right onto FM 3232.